Excluding Antarctica, the two closest points (by a large factor) on land which are on opposite sides of the date line are the Diomede Islands, which are about 4km apart. The International Date Line falls neatly between the western island (Big Diomede), which is part of Russia and uses UTC+12, and the eastern island (Little Diomede), which is in the US and uses UTC-9. These islands are sometimes known as “tomorrow island” and “yesterday island”.
Little Diomede has a little over 100 permanent inhabitants and can be reached by helicopter or by airplane in the winter. There are no formal hospitality structures — this isn't a place where people go to party for the new year or otherwise. Big Diomede is home to a Russian military base and you cannot visit it under normal circumstances. In theory, I think you can walk between the two islands (December temperature still rarely go above -25°C = -12°F, so the ice shelf should still extend that far); however the Russian authorities are likely to object.
Anywhere else in the world, traveling across the IDL involves flying (or sailing, but it won't be fast enough) between archipelagos. Likely journeys are Tonga (UTC+13) to Niue (UTC-11) (about 600km), Apia, Samoa (UTC+14) to Niue (UTC-11) (about 600km) or Kirimati, Kiribati (UTC+14) to Tahiti, French Polynesia (UTC-10) (about 2300km). I don't know if there are any suitable scheduled flights.
Another option would be to sail on January 1 in a ship from a place that's in the western hemisphere but west of the IDL, such as most of Kiribati: when you exit the territorial waters, the time switches from the local time of the adjoining land to the time determined by the longitude rounded to the nearest 15°. Conversely, you can arrange to be outside of territorial waters on December 1 and reach a place that's in the eastern hemisphere but east of the IDL; this only includes US islands, none inhabited except for a few in the Aleutians.